Chapter 9: Non-parametric Tests
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Parametric vs Non-parametric
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Chi-Square
– 1 way
– 2 way
Parametric Tests
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Data approximately normally distributed.
Dependent variables at interval level.
Sampling random
t - tests
ANOVA
Non-parametric Tests
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Do not require normality
Or interval level of measurement
Less Powerful -- probability of rejecting
the null hypothesis correctly is lower. So
use Parametric Tests if the data meets
those requirements.
One-Way Chi Square Test

Compares observed frequencies within
groups to their expected frequencies.

HO = “observed” frequencies are not
different from the “expected”
frequencies.
Research hypothesis: They are
different.

Chi Square Statistic
 fo = observed frequency
 fe = expected frequency
Chi Square Statistic

2
( fo  fe)

fe
2
One-way Chi Square

Calculate the Chi Square statistic
across all the categories.

Degrees of freedom = k - 1, where k is
the number of categories.
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Compare value to Table of Χ2.
One-way Chi Square
Interpretation

If our calculated value of chi square is less
than the table value, accept or retain Ho

If our calculated chi square is greater than the
table value, reject Ho
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…as with t-tests and ANOVA – all work on the
same principle for acceptance and rejection
of the null hypothesis
Two-Way Chi Square
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Review cross-tabulations (=
contingency tables) from Chapter 2.
Are the differences in responses of two
groups statistically significantly
different?
One-way = observed vs expected
Two-way = one set of observed
frequencies vs another set.
Two-way Chi Square
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Comparisons between frequencies
(rather than scores as in t or F tests).
So, null hypothesis is that the two or
more populations do not differ with
respect to frequency of occurrence.
rather than working with the means as
in t test, etc.
Two-way Chi Square Example
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Null hypothesis: The relative frequency
[or percentage] of liberals who are
permissive is the same as the relative
frequency of conservatives who are
permissive.
Categories (independent variable) are
liberals and conservatives. Dependent
variable being measured is
permissiveness.
Two-Way Chi Square Example
Child-rearing
Practices
Permissive
Non-permissive
Total
Political
Orientation
Liberals
13
7
20
Conservatives
7
13
20
Total
20
20
40
Two-Way Chi Square Example
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Because we had 20 respondents in
each column and each row, our
expected values in this cross-tabulation
would be 10 cases per cell.
Note that both rows and columns are
nominal data -- which could not be
handled by t test or ANOVA. Here the
numbers are frequencies, not an
interval variable.
Two-Way Chi Square Expected
Child-rearing
Practices
Permissive
Non-permissive
Total
Political
Liberals
10
10
20
Orientation (Expected)
Conservatives
10
10
20
Total
20
20
40
Two-Way Chi Square Example
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Unfortunately, most examples do not
have equal row and column totals, so it
is harder to figure out the expected
frequencies.
Two-Way Chi Square Example
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What frequencies would we see if there
were no difference between groups (if
the null hypothesis were true)?
If 25 out of 40 respondents(62.5%) were
permissive, and there were no
difference between liberals and
conservatives, 62.5% of each would be
permissive.
Two-Way Chi Square Example
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We get the expected frequencies for
each cell by multiplying the row
marginal total by the column marginal
total and dividing the result by N.
We’ll put the expected values in
parentheses.
Two-Way Chi-Square Example
Political Orientation
Permissive
Not Permissive
Liberals Conservatives Total
15 (12.5) 10 (12.5)
25
5 (7.5)
10 (7.5)
15
Total
20
20
40
Two-Way Chi-Square Example
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So the chi square statistic, from this
data is
(15-12.5)squared / 12.5 PLUS the same
values for all the other cells
= .5 + .5 + .83 + .83 = 2.66
Two-Way Chi-Square Example

df = (r-1) (c-1) , where r = rows, c
=columns so df = (2-1)(2-1) = 1
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From Table C, α = .05, chi-sq = 3.84
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Compare: Calculate 2.66 is less than
table value, so we retain the null
hypothesis.
Chapter 9: Non-parametric Tests
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Review Parametric vs Non-parametric
Be able to calculate:
Chi-Square (obs-exp2 ) / exp
– 1 way
– 2 way
• (row total) x (column total) / N = expected
value for that cell
• calculate chi-square and compare to table.